equivalent conditions for uniform integrability
For every there is a so that
for all and .
For every there is a satisfying
for all .
So, for bounded subsets of , either of the above properties can be used to define uniform integrability. Conversely, when the measure space is finite, then conditions (2) and (3) are easily shown to imply that is bounded in .
To show the equivalence of these statements, let us suppose that for .
and, therefore, .
For each , property (2) gives a satisfying
First, suppose that for . For , the condition that as gives a such that whenever . Setting ,
whenever and .
|Title||equivalent conditions for uniform integrability|
|Date of creation||2013-03-22 18:40:17|
|Last modified on||2013-03-22 18:40:17|
|Last modified by||gel (22282)|